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This text is based on a lecture course given by the authors in the framework of Oberwolfach Seminars at the Mathematisches Forschungsinstitut Oberwolfach in May, 2013. It is intended to serve as a thorough introduction to the rapidly developing field of positional games. This area constitutes an important branch of combinatorics, whose aim it is to systematically develop an extensive mathematical basis for a variety of two player perfect information games. These ranges from such popular games as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. The subject of positional games is strongly related to several other branches of combinatorics such as Ramsey theory, extremal graph and set theory, and the probabilistic method. These notes cover a variety of topics in positional games, including both classical results and recent important developments. They are presented in an accessible way and are accompanied by exercises of varying difficulty, helping the reader to better understand the theory. The text will benefit both researchers and graduate students in combinatorics and adjacent fields.

The seminar focuses on a recent solution, by the authors, of a long standing problem concerning the stable module category (of not necessarily finite dimensional representations) of a finite group. The proof draws on ideas from commutative algebra, cohomology of groups, and stable homotopy theory. The unifying theme is a notion of support which provides a geometric approach for studying various algebraic structures. The prototype for this has been Daniel Quillen's description of the algebraic variety corresponding to the cohomology ring of a finite group, based on which Jon Carlson introduced support varieties for modular representations. This has made it possible to apply methods of algebraic geometry to obtain representation theoretic information. Their work has inspired the development of analogous theories in various contexts, notably modules over commutative complete intersection rings and over cocommutative Hopf algebras. One of the threads in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This story started with Mike Hopkins' classification of thick subcategories of the perfect complexes over a commutative Noetherian ring, followed by a classification of localizing subcategories of its full derived category, due to Amnon Neeman. The authors have been developing an approach to address such classification problems, based on a construction of local cohomology functors and support for triangulated categories with ring of operators. The book serves as an introduction to this circle of ideas.

<p>This book concentrates on the mathematics of photonic crystals, which form an important class of physical structures investigated in nanotechnology. Photonic crystals are materials which are composed of two or more different dielectrics or metals, and which exhibit a spatially periodic structure, typically at the length scale of hundred nanometers.</p><p>In the mathematical analysis and the numerical simulation of the partial differential equations describing nanostructures, several mathematical difficulties arise, e. g., the appropriate treatment of nonlinearities, simultaneous occurrence of continuous and discrete spectrum, multiple scales in space and time, and the ill-posedness of these problems.</p><p>This volume collects a series of lectures which introduce into the mathematical background needed for the modeling and simulation of light, in particular in periodic media, and for its applications in optical devices.</p>

This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature.

This book is devoted to the study of coupled partial differential equation models, which describe complex dynamical systems occurring in modern scientific applications such as fluid/flow-structure interactions.

This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples.

The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation.

These notes are based on lectures presented during the seminar on " Asymptotic Statistics" held at SchloB Reisensburg, Gunzburg, May 29-June 5, 1988. They consist of two parts, the theory of asymptotic expansions in statistics and probabilistic aspects of the asymptotic distribution theory in nonparametric statistics. Our intention is to provide a comprehensive presentation of these two subjects, leading from elementary facts to the advanced theory and recent results. Prospects for further research are also included. We would like to thank all participants for their stimulating discussions and their interest in the subjects, which made lecturing very pleasant. Special thanks are due H. Zimmer for her excellent typing. We would also like to take this opportunity to to express our thanks to the Gesellschaft fur mathematische Forschung and to the Deutsche Mathematiker Vereinigung, especially to Professor G. Fischer, for the opportunity to present these lectures and to the Birkhauser Verlag for the publication of these lecture notes. R. Bhattacharya, M. Denker Part I: Asymptotic Expansions in Statistics Rabi Bhattacharya 11 §1. CRAMER-EDGEWORTH EXPANSIONS Let Q be a probability measure on (IRk, B"), B" denoting the Borel sigmafield on IR". Assume that the s - th absolute moment of Q is finite, (1.1) P. := J II x lis Q(dx) < 00, for some integer s;::: 3, and that Q is normalized, (1.2) J x(i)Q(dx) = 0 (1 ~ i ~ k), J x(i)x(j)Q(dx) = Dij (1 ~ i,j ~ k).

They are related to representations of finite groups and group actions on spheres, and are considered as a generalisation of the spherical space form problem. The second part reviews the general setting of surgery theory and reports on the computation of the surgery abstraction groups.

thus the connectedness theorem can be used to prove the inequality-part of Hartshorne's conjecture on linear normality, whereas Deligne's generalisation of the connectedness theorem leads to a refinement of Barth's results on the topology of varieties with small codimension in a projective space.

This book surveys research results on the physical and mathematical modeling, as well as the numerical simulation of complex fluid and structural mechanical processes occurring in the human blood circulation system. choice of suitable liquid and wall models; mathematical analysis of coupled models; mathematical analysis of piping systems;

These notes are based on lectures given in the semmar on "Coding Theory and Algebraic Geometry" held at Schloss Mickeln, Diisseldorf, November 16-21, 1987. In 1982 Tsfasman, Vladut and Zink, using algebraic geometry and ideas of Goppa, constructed a seqeunce of codes that exceed the Gilbert-Varshamov bound. The result was considered sensational. Furthermore, it was surprising to see these unrelated areas of mathematics collaborating. The aim of this course is to give an introduction to coding theory and to sketch the ideas of algebraic geometry that led to the new result. Finally, a number of applications of these methods of algebraic geometry to coding theory are given. Since this is a new area, there are presently no references where one can find a more extensive treatment of all the material. However, both for algebraic geometry and for coding theory excellent textbooks are available. The combination ofthe two subjects can only be found in a number ofsurvey papers. A book by C. Moreno with a complete treatment of this area is in preparation. We hope that these notes will stimulate further research and collaboration of algebraic geometers and coding theorists. G. van der Geer, J.H. van Lint Introduction to CodingTheory and Algebraic Geometry PartI -- CodingTheory Jacobus H. vanLint 11 1. Finite fields In this chapter we collect (without proof) the facts from the theory of finite fields that we shall need in this course.

Der. vorliegende Band enthält eine Reihe von einführenden Vorlesungen, die von verschiedenen Autoren im Rahmen von zwei DMV-Seminaren zum Thema "Algebraische Transjormationsgruppen und Invariantentheorie" gehalten wur­ den. Entsprechend der allgemeinen Zielsetzung der DMV-Seminare sollten sowohl grundlegende Techniken und Resultate vorgestellt als auch Einblicke in aktuelle Entwickl~ngen gegeben werden. Was die Grundlagen anbetrifft, so haben wir sie hier nicht in vollem Umfang widergegeben. Im Bedarfsfall mag der Leser unsere Bücher "Geometrische Methoden in der Invariantentheorie"l und "Invariant Theory"2 zu Rate ziehen, auf die sich die einführenden Vorträge stützten. Leider konnten auch nicht alle aktuellen Entwicklungen berücksichtigt werden, über die im Seminar berichtet wurde. Die Ziele der hier vorliegenden Beiträge, auf deren Inhalt wir in der Einführung ausführlicher eingehen werden, sind entsprechend unterschiedlicher Natur. Einige liefern Darstellungen bereits publizierter Theorien, wobei sie allerdings ein größeres Gewicht auf Motivation und die Ausführung von Beispie­ len legen, als dies in den Originalarbeiten möglich war. Andere leiten grundle­ gende Resultate auf neue "reise her oder stellen sie aus anderer Sicht dar. Schließlich werden auch noch einzelne Einblicke in aktuelle Forschungsrichtun­ gen gegeben. Wir hoffen, daß durch diesen Band zahlreiche Resultate der Theorie der algebraischen Transformationsgruppen leichter zugänglich geworden sind, und daß der Leser mit ihm eine nützliche Basis für die Lektüre aktueller Forschungsarbeiten erhält.

The following notes grew out oflectures held during the DMV-Seminar on Random Media in November 1999 at the Mathematics Research Institute of Oberwolfach, and in February-March 2000 at the Ecole Normale Superieure in Paris.

These notes are based on lectures presented during the seminar on " Asymptotic Statistics" held at SchloB Reisensburg, Gunzburg, May 29-June 5, 1988. They consist of two parts, the theory of asymptotic expansions in statistics and probabilistic aspects of the asymptotic distribution theory in nonparametric statistics. Our intention is to provide a comprehensive presentation of these two subjects, leading from elementary facts to the advanced theory and recent results. Prospects for further research are also included. We would like to thank all participants for their stimulating discussions and their interest in the subjects, which made lecturing very pleasant. Special thanks are due H. Zimmer for her excellent typing. We would also like to take this opportunity to to express our thanks to the Gesellschaft fur mathematische Forschung and to the Deutsche Mathematiker Vereinigung, especially to Professor G. Fischer, for the opportunity to present these lectures and to the Birkhauser Verlag for the publication of these lecture notes. R. Bhattacharya, M. Denker Part I: Asymptotic Expansions in Statistics Rabi Bhattacharya 11 §1. CRAMER-EDGEWORTH EXPANSIONS Let Q be a probability measure on (IRk, B"), B" denoting the Borel sigmafield on IR". Assume that the s - th absolute moment of Q is finite, (1.1) P. := J II x lis Q(dx) < 00, for some integer s;::: 3, and that Q is normalized, (1.2) J x(i)Q(dx) = 0 (1 ~ i ~ k), J x(i)x(j)Q(dx) = Dij (1 ~ i,j ~ k).

Infinite dimensional manifolds, Lie groups and algebras arise naturally in many areas of mathematics and physics.

These notes form an extended version of talks given at the DMV-seminar in Giinzburg, September 1990. The seminar consisted of two parts: 1) "The isomorphism problem for integral group rings", with the main talks given by K. W. Roggenkamp and shorter contributions by W.Kimmerle, J.llitter and A. Zimmermann (Part 1). 2) "Galois-Module structure", with the main talks given by M.Taylor and shorter contributions by N.Byott (Part 2). We greatly appreciate the opportunity, given us by the DMV to hold this seminar. DMV-Seminar Part 1 Group Rings: Units and the Isomorphism Problem K. W. Roggenkamp with contributions by W. Kimmerle and A. Zimmermann Contents 3 Table of Contents I Some general facts ... 7 § 1 Ring reduction to PID . 7 § 2 Modules ........ . 7 II Some notes on representation theory 9 §1 Blocks .................. . 9 §2 Normalizers of p-Sylow subgroups of group bases 11 §3 Cohomology rings 12 Profinite groups .. . . . . . . . . 13 §4 III The leading coefficient of units 15 IV Class sum correspondence ... . . . . . . . . . . . .. . . 21 .

One of the key concerns in modern control theory is the design of steering strategies. The implementation of such strategies is done by a regulator. Presented here is a self-contained introduction to the mathematical background of this type of regulator design. The topics selected address the matter of greatest interest to the control community, at present, namely, when the design objective is the reduction of the influence of exogeneous disturbances upon the output of the system. In a first scenario the disturbance signal is regarded as a deterministic time series with known dynamics but unknown parameters. The design objective is then the asymptotic disturbance compensation. In a second scenario, no information about the disturbance signal is available apart from some bounds. Here, in an H-approach, control strategies are worked out which will prove efficient for all such disturbances. The intention of this book is to present ideas and methods on such a level that the beginning graduate student will be able to follow current research. New results are included, especially for nonlinear control systems, and as a service to the reader, an extensive appendix presents topics from linear algebra, invariant manifolds and calculus of variations, information which is hardly to be found in standard textbooks. Contents: Introduction ¿ The problem of output regulation ¿ Introduction ¿ Problem statement ¿ Output regulation via full information ¿ Output regulation via full error feedback ¿ A particular case ¿ Well-posedness and robustness ¿ The construction of a robust regulator ¿ Disturbance attenuation via H-methods ¿ Introduction ¿ Problem statement ¿ A characterization of the L2-gain of a linear system ¿ Disturbance attenuation via full information ¿ Disturbance attenuation via measured feedback ¿ Full information regulators ¿ Problem statement ¿ Time-dependent control strategies ¿ Examples ¿ Time-independent control strategies¿ The local case ¿ Nonlinear observers ¿ Problem statement ¿ Time-dependent observers ¿ Error feedback regulators ¿ Examples ¿ Nonlinear H-techniques ¿ Introduction ¿ Construction of the saddle-point ¿ The local scenario ¿ Disturbance attenuation via linearization ¿ Matrix equations ¿ Linear matrix equations ¿ Algebraic Riccati equations ¿ Invariant manifolds ¿ Existence theorem ¿ Outflowing manifolds ¿ Asymptotic phase ¿ Convergence for T ¹ ¿ A special case ¿ Dichotomies and Lyapunov functions ¿ Hamilton-Jacobi-Bellman-Isaacs equation ¿ Introduction ¿ Method of characteristics ¿ The equation of Isaacs ¿ The Hamiltonian version of Isaacs' equation ¿ Bibliography

Questions that arose from linear programming and combinatorial optimization have been a driving force for modern polytope theory, such as the diameter questions motivated by the desire to understand the complexity of the simplex algorithm, or the need to study facets for use in cutting plane procedures.

PHug) 7 Markovian stochastic optimization and stochastic approximation procedures 53 8 Asymptotic distributions 71 9 Stopping times 79 1O Applications of stochastic approximation methods 80 References for Part II 90 III Applications to adaptation algorithms (L.

Conformal invariants (conformally invariant tensors, conformally covariant differential operators, conformal holonomy groups etc.) are of central significance in differential geometry and physics.

Specifically, Part 1 treats the deformation theory, existence and geometry of rational curves via characteristic p, while Part 2 is principally concerned with vanishing theorems and their geometric applications.

The book provides apedagogical entry into an active area of ongoing research for bothgraduate students and researchers. It is an outgrowth of a coursegiven by the authors for graduate students and post-doctoralresearchers at the Oberwolfach Research Institute in 2004.

Topological K-theory is one of the most important invariants for noncommutative algebras. The book studies a number of applications, including K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem.

Higher Dimensional Algebraic Geometry presents recent advances in the classification of complex projective varieties. Recent results in the minimal model program are discussed, and an introduction to the theory of moduli spaces is presented.

Takes Weyl's "Raum - Zeit - Materie" (Space - Time - Matter) as center of concentration and starting field for a look at his work. This book discusses Weyl's deep involvement in relativity, cosmology and matter theories between the classical unified field theories and quantum physics.

How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques.

These notes present a polished introduction to tropical geometry and contain some applications of this rapidly developing and attractive subject. It consists of three chapters which complete each other and give a possibility for non-specialists to make the first steps in the subject which is not yet well represented in the literature.

Our DMV Seminar on 'Classical Nonintegrability, Quantum Chaos' intended to introduce students and beginning researchers to the techniques applied in nonin tegrable classical and quantum dynamics.